In 1990, the "Mendelssohn" Stradivarius violin sold at Christie's in London for $ 1,686,700. A good violin at a typical music store sells for around $ 2,000. What is it about the Stradivarius that makes it cost almost 1,000 times as much? The structure and geometry of the two instruments are very similar, yet subtle differences in the structural dynamics of the two instruments cause them to vibrate differently in response to an excitation by a violinist's bow. This, in turn, causes differences in the sound produced by the two instruments which ultimately determines quality and, to a large extent, price. If it were possible to force the less expensive violin to vibrate like the Stradivarius, the legendary sound would follow.
The relatively new field of smart structural/acoustic control is centered around changing the structural dynamics of an acoustically radiative structure to change, usually to suppress, the sound resulting from vibration of the structure. This is done by connecting actuators that are integrated into the structure in a control loop with sensors that are either in the acoustic field or also integrated in the structure. Smart structural/acoustic control also has the potential to force one acoustically radiative structure to behave like a target acoustically radiative structure, thus replicating its acoustic properties. The less expensive violin might be forced to sound like a Stradivarius. The concept of acoustic replication using smart structures has far reaching implications, from the field of acoustic musical instruments to aircraft cockpits.
To provide a background, a brief review of active acoustics leading to smart structural acoustics is presented. Smart structural acoustics is a relatively recent subset of the broader field of acoustic control wherein an acoustically noisy structure may be controlled at the structure through integrated sensors and actuators. This integration is such that the sensors and actuators are load-carrying parts of the structure as well as control elements. The field of smart structural acoustics has emerged in a natural progression: first, acoustic control by acoustic sources; then, by vibration inputs; and finally, by integrated sensors and actuators or, smart structural acoustic control.
Additionally, a review is given of literature on the acoustic guitar. This instrument has inspired a significant amount of analytical and experimental research from the perspective of acoustics and structural dynamics. As such, there are identified dynamic parameters in the literature that could potentially be further "tuned" using active acoustic control to accomplish desired changes in acoustic parameters.
In most applications, acoustic control is implemented in order to suppress unwanted noise through attenuation or other mechanisms. Sound attenuation is usually implemented through sound-absorbing materials for sounds of medium and high frequencies. Because the thickness of the sound absorption material necessary to produce constant attenuation increases with decreasing frequencies, there is a practical limit on its use at relatively low frequencies. In this low frequency region, active acoustic control has found applications.
The principles underlying active acoustic control have been understood at least since 1802 when Young's principle of interference was introduced. The principle suggests cancellation of a sound wave propagating in space by the addition of an inverse wave. This principle forms the basis of active noise control. Huygen's principle, as applied to acoustics, is an extension of Young's principle for multiple dimensions. Huygen's principle states that the sound field inside a surface that is produced by a source outside the surface can be exactly reproduced by an infinite array of secondary sources distributed along the surface. Since an infinite array of secondary sources are not realizable, in practice, a finite number of secondary sources can be "field-fitted" to achieve an optimum result.
Despite the longevity of the underlying principles of active noise control, one of the first practical implementations was described by Lueg in a German patent in 1933 and in a U.S. patent in 1934 (U.S. Pat. No. 2,043,416). Phase reversal in Lueg's one-dimensional duct was accomplished by considering the electronic system as a transmission line whose length determined the time delay. Lueg also proposed cancellation in a space very near a loudspeaker and in an open space using a microphone and a loudspeaker. It has been found more recently that cancellation at a point is done at the expense of increased noise at other locations in the field. Also, Lueg's approach to control of noise in an open space was probably not viable since successful experiment implementations of this are much more recent and inevitably involve more than one microphone and speaker.
Little was published in the field of active control following Lueg's patent until the 1950's. In 1953, Olson published research on an electronic sound absorber and Conover made early attempts to control transformer noise using a single loudspeaker. Frequency performance range of Olson's devices were limited at low frequencies by loudspeaker performance and at high frequencies by phase errors and electronics. An attenuation is achieved of almost 25 dB in the range of 60 to 80 Hz accompanied by an almost linearly decreasing attenuation up to around 500 Hz where there is an increase of sound pressure of 5 dB. This early work started to map out the frequency range of usefulness of active versus passive noise control, where active is most effective in the range of near DC to 500 Hz and passive is most effective above 500 Hz. This upper limit on active control should continue to increase as theory develops, computing power continues to increase, and computing equipment cost continues to decrease.
Applications in which modern active noise control research continue are plentiful, including approximately one-dimensional problems such as ducts and noise-reducing headsets and multidimensional applications such as cylinder interiors and transformers. Cylinder interiors are of particular interest because of their natural extension to fuselages and launch vehicles.
The idea of noise reducing headsets started as a more advanced version of Lueg's system for controlling duct noise and was implemented by Olson. For low frequencies, sound waves in ducts propagate as approximately one-dimensional plane waves. As the sound frequency increases, the sound propagation becomes multidimensional and much harder to control as the plane wave assumption breaks down and transverse resonances cause pressure fluctuations through a cross section. Active noise control has been applied to fan-induced duct noise in commercial air handlers at low frequencies. The limiting frequency for noise reduction of up to 20 dB for most duct structures is around 500 Hz. This limitation is also imposed by sampling and processing speeds.
Internal cylinder noise can be a pseudo two-dimensional problem or a three-dimensional problem depending on whether the noise sources and secondary sources lie in the same cross-sectional plane and the frequency of the noise. In 1976, Kempton, put forth one of the first illustrations of a multidimensional active acoustic control problem using an array of "anti-sources" to cancel the far-field of a monopole source. Lester and Fuller used four interior monopole control sources to attenuate noise by around 20 dB within a cylindrical cross section caused by 2 exterior monopole noise sources. Later, Fuller, and Jones and Jones and Fuller performed similar studies using a structural control actuator. These will be covered in greater detail in the next section. Elliot et al. determined that as long as secondary sources couple sufficiently with modes that are excited by the primary source, it is possible to achieve noise reduction without locating secondary sources near the primary source. Noise control has also been applied to the characteristic low frequency hum of transformers. Angevine showed attenuation levels of 16 dB using 26 secondary sources surrounding the transformer.
When the source of noise to be controlled is a structure, the use of acoustic sources for control is available in addition to the option of applying a vibrational source directly to the structure. The addition of a sensor and a control methodology can potentially modify the structure so that noise does not propagate as readily at the frequencies of interest. An advantage for direct structural actuators is illustrated by an inherent disadvantage in acoustic source control. When there are many phase changes across the surface of a noise source, as in a panel structure vibrating in a higher mode, many acoustic sources are needed for control. In the case of the panel, there should be at least one acoustic source for each antinode on the structure. Additionally, it has been found in the control of interior noise of cylinders that direct structural actuation avoids control spillover effects encountered using acoustic sources. Control spillover is the effect of generating additional, unwanted noise when control is implemented due to an inexact match of the control field to the primary field with respect to spatial distribution.
Some of the earliest works in the literature involving direct structural actuation to provide vibration inputs were published in the Soviet Union. In 1966, Knyazev and Tartakovskii used vibration pickups and vibration inputs to control plate vibrations by introducing active damping. They also noticed an average reduction of 16 dB in acoustic pressure over the area of the plate when vibrating at 390 Hz. This frequency was located very close to a resonance of the plate. A follow-up paper in 1967, by Knyazev and Tartakovskii, was directed primarily at acoustic attenuation of noise radiated by the flexural waves of a plate. Experimental results indicated an average of 7 dB reduction in acoustic pressure across a frequency range of DC to 1900 Hz. They noted that the tuning of vibration dampers to minimize the noise field does not coincide with the tuning of vibration dampers to minimize vibration and that the maximum radiation attenuation of noise occurs near the location of the damper. In another relatively early publication from the Soviet Union in 1987, Vyalyshev, Dubinin, and Tartakovskii presented a theoretical examination of reductions in sound transmission through a plate with an auxiliary point force used as a control actuator. They observed that reductions in sound transmission through the plate could alternately be viewed as an increase in the impedance of the plate.
Early pioneering work in the United States using direct structural actuators to provide vibration inputs began with Jones and Fuller on active control of a sound field within a cylinder (this followed an earlier reference work by Lester and Fuller using acoustic sources on the same problem). This cylinder study was directed towards the control of cabin noise in the advanced turboprop aircraft. A control relation is derived, in this experimental study, by producing the same sound field at a given microphone location using both an acoustic source that is supposed to simulate noise and a secondary vibration control source. Both sources were then switched on and their phase varied with respect to each other while sound pressure level (SPL) was measured at several interior locations as a function of this variation. Both resonant and off-resonant noise frequencies were investigated. Attenuation of sound pressure of up to 20 dB was obtained. An additional study by Jones and Fuller showed reductions of up to 30 dB at acoustic resonance in the cavity using two vibration control sources and two microphone error sensors. In this case, the control was formulated by minimizing a quadratic cost function based on error signals from the microphones.
An enhancement to providing direct structural actuation with a point force is to provide direct structural actuation using actuators that have been developed for smart structures. The use of smart structures started in the field of vibration control. In acoustic control, the objective changes from one of minimizing or altering structural response to one of minimizing or altering acoustic response. These two objectives often require very different control laws, but both may be achievable using the same actuator. A smart structure actuator can either be imbedded in or bonded to the host structure. It provides a source of direct structural actuation without the added space and structural grounding requirements necessary with a shaker providing a point force. In addition, point force actuation is more prone to spillover and shakers exhibit a certain back reactance that may require consideration in the model of the structure. Smart structure actuators only slightly increase the mass and stiffness at the point of application. The primary smart structure actuator used, in vibration applications, is the surface-bonded piezoceramic. Transverse deflections on application of a voltage in the poling direction of the through-the-thickness poled piezoceramic translate into in-plane surface tractions applied to the structure.
The first investigation of what could be called a smart structure actuator was directed at vibration control by Forward. He used bonded piezoceramics as sensors and actuators to control the vibration of a mirror subjected to acoustic excitation. Other early work, which concentrated on vibration control of beam structures, includes that of Bailey and Hubbard, who investigated the use of poly vinyldene fluoride (PVDF), a piezoelectric polymer, as a distributed parameter actuator on a cantilever beam. Obal and Hanagud, Obal, and Calise formulated an optimal control law for vibration suppression of a beam using surface-bonded piezoceramic sensors and actuators. They also found that for the assumptions of uniform beam stiffness and perfectly rigid bonds, piezoceramics could be modeled as concentrated line moments applied to the beam at the boundaries of the actuators. Baz and Poh investigated optimal location and control gains for minimizing beam vibration amplitude using piezoceramic actuators. The interaction between piezoceramic actuators and beam structures was first thoroughly analyzed by Crawley and De Luis and later by Crawley and Anderson. An important conclusion was that the bonding layer should be very thin and that the piezoceramic actuator should be stiff compared to the host structure for maximum force at a given voltage. They also came to a similar conclusion as references to Obal and Hanagud, et al. that, under these conditions, the action on the beam by the piezoceramic can be approximated by line moments proportional to the applied voltage at the boundaries of the piezoceramic. Early work on the incorporation of one-dimensional active piezoceramic elements into more complicated truss structures for vibration suppression was done by Fanson and Chen. More recently, Bronowicki and Betros developed a hybrid method for modeling piezoceramic sensing and actuation of complicated truss-beam combination structures which uses a finite element code to generate structural mode shapes and a thermal analogy to model both sensing and actuation.
Investigations into the more general problem of actuation of plates using surface-bonded piezoceramic actuators are more relevant to acoustic problems, but also have their background in vibration suppression problems. Approaches to smart structure plate actuation can be divided into two categories: (1) continuous exact or approximate solutions and (2) discrete formulations involving a finite element model (FEM).
Among continuous solutions, Dimitriadis, Fuller, and Rogers put forward a theoretical paper postulating the interaction between a piezoceramic plate bonded to a plate substructure. A perfect bond and a uniform bending applied by the actuator at all points within the actuator boundaries were assumed, resulting in a spherical deformation of the plate due to the actuator. It was predicted, analogous to the beam case, that the piezoceramic could be replaced by line moments along the borders of the piezoceramic actuator. Also, it was shown that for symmetric distribution of an actuator about a nodal line of a given vibrational mode, excitation of that mode was theoretically impossible. Optimum actuator position for excitation of a vibrational mode was said to be near nodal lines. A more general statement of this principle by Fuller, Rogers, and Robertshaw is that the center of the actuator should be in a region of high structural surface strain of a mode for excitation of that mode. Crawley and Lazarus developed a model of induced strain actuation that was applicable to isotropic and anisotropic plates. The model was experimentally verified for the case of piezoceramic material covering the majority of both surfaces of cantilevered plate test articles in static deflection due to voltage applied to the actuators. Kim and Jones included the effect of a finite thickness bonding layer in actuation of a plate by surface-bonded piezoceramic actuators. They also presented some results on optimal thicknesses of the actuator for a constant applied field. In a study of segmentation of piezoceramic sensors and actuators bonded onto plates, Tzou and Fu found that proper segmentation of piezoceramics result in the ability to sense and actuate modes for which piezoceramics are evenly distributed about a nodal line of the mode.
The inherent limitation in all of the continuous models is that the plate substructure problem must be amenable to a continuous exact or approximate solution in order to solve the combined piezoceramic/plate problem. For evaluation of potentially more complex problems, approaches have been developed which fall into the category of discrete solutions involving FEM. The first piezoelectric finite element for structural dynamics that could be found was derived by Allik and Hughes. Also, McDearmon published a method to add piezoelectric properties to structural finite elements through a matrix manipulation of elastic and heat transfer element matrices. In a much more recent study, Ha, Keilers, and Chang developed a composite finite element with piezoceramics included as outer layers of the element. The specific element was eight-noded, with three displacement degrees of freedom and one voltage degree of freedom per node. A modal expansion was used to show the feasibility of introducing active damping although no explicit control algorithm was formulated. Comparisons were also made between predictions of static and dynamic deflections using an assembled model that included the composite element and experimental data on cantilevered plates.
Piezoceramics are also used as actuators in the majority of smart structure acoustic control research found. Piezoceramics offer the necessary frequency response and force authority for active acoustic control. In addition, the distributed nature of the piezoceramic wafer can be used to spatially filter selected modes that are acoustic radiators by proper placement of the actuator material. Rogers, Fuller, and Liang have also proposed using embedded nitinol fibers, a shape memory alloy, to control sound transmission through a panel. Activation of the nitinol fibers results in a static change in mechanical properties and mode shapes of the panel that can reduce sound transmission.
There have been a number of theoretical papers considering smart structural acoustic control applied to both beam and plate structures. Clark and Gibbs investigated the use of a simply supported plate with one piezoceramic actuator to demonstrate a higher harmonic control approach. Control of sound radiation due to subsonic vibrational waves impinging on structural discontinuities was researched by Guigou and Fuller. In this study, active control forces due to bonded piezoceramics and shakers, were both shown to be effective at minimizing the radiated acoustic field. Clark and Fuller present a theoretical paper examining model reference-based control on the acoustic field resulting from a simply supported beam with piezoceramic actuators and structural sensors. The structural response is driven by a controller to some predetermined reference response which results in favorable acoustic response. It was shown analytically that the same degree of control that can be achieved by any number of error sensors in the acoustic field and n actuators can also be achieved by using n structural sensors and n actuators. This provides a means to get a high degree of acoustic control through a detailed initial survey using many microphones in the acoustic field, and to maintain that control with a reduced number of structural sensors.
There have also been studies that include experimental validation implementing smart structural acoustic control of plates. In a purely experimental study, Fuller, Hansen, and Snyder achieve a global attenuation on the order of 45 dB using a piezoceramic actuator and a form of open-loop control which varies the phase between the disturbance and the control signals. This was done at two distinct resonant frequencies of a simply supported plate. In another experiment, Clark and Fuller compare the number of piezoceramic actuators used to control on-resonant and off-resonant excitation of a simply-supported plate. They found that for on-resonant excitation, more piezoceramic actuators failed to elicit better performance, while for off-resonant cases more piezoceramic actuators increased performance. Also, Clark and Fuller give an optimal placement methodology for piezoceramic actuators and PVDF structural sensors on a baffled, simply-supported plate. A Rayleigh integral approach is used to predict pressure fluctuation as a result of plate movement. Analytical results formulated using a linear quadratic optimal control theory are compared to experimental results. It was found that a single optimally-placed piezoceramic actuator and PVDF sensor can rival performance achieved with three arbitrarily-placed actuators and three microphone sensors. Van Niekerk, Tongue, and Packard used a pair of surface-bonded piezoceramic actuators mounted on a circular plate that was mounted in a duct to suppress a transient pressure pulse due to a loudspeaker that was also mounted in the duct. They found reductions of up to 15 dB in a microphone that was placed downstream of the plate when the controller was active.
Smart structural acoustic control applied to flexible plates that are backed by sealed rigid cavities has also been the subject of a small body of recent research. This model is important because it adds insight to problems of sound propagation into aircraft cabins, where the primary noise source is due to new, more efficient, but noisier turboprop engines and into spacecraft launch vehicles where excitation of the payload fairing can create a harsh enough internal acoustic field to interfere with sensitive payloads. Lyon was the first reference found to investigate passive suppression of sound propagation into a sealed, cavity-backed plate, but the first references investigating smart structural acoustic control on the related problem of sound propagation into a two-dimensional cavity with a flexible beam boundary were by Banks and Fang almost 30 years later, in 1991. In this later theoretical work, piezoceramic actuators were bonded to both sides of a clamped, flexible beam boundary, and a time domain state space formulation was derived for coupled structure/fluid system and used to investigate active control of noise in the cavity and beam amplitude due to a periodic beam excitation. Kohsigoe, Gillis, and Falangas investigated sound transmission through an elastic, simply-supported plate into a three-dimensional cavity with rigid sides, a lightly damped back wall, and a rigid inner box located at the center of the cavity. The theoretical development includes a formulation for the equation of motion of the plate and equations for resulting pressure inside and outside of the cavity. Active noise control is investigated for controlling noise transmission into the cavity using the piezoceramics as actuators. In an entirely experimental study, Ellis and Koshigoe constructed a cavity with rigid sides and back and clamped a flexible plate to the front with a piezoceramic actuator and accelerometer sensor in order to study control of harmonic noise transmission due to an external loudspeaker. In a theoretical study, Koshigoe and Ellis considered decreasing harmonic noise transmission through a simply-supported plate with surface-bonded piezoceramic actuators into a rigid cavity with a time-varying mean air density. Hill et al. conducted an experimental investigation of decreasing harmonic sound transmission due to a loudspeaker through a clamped plate with a pair of surface bonded piezoceramic actuators into a sealed, rectangular cavity with acoustically reflective sides and back. Low-order models, which captured the modes to be controlled, were fit to measured data for state space control design.
Two approaches are available for sensing in acoustic control of structures. The traditional approach is to sense the acoustically radiated field directly using microphones in the acoustic field. The second approach is to use any one of the smart structural sensors that have been developed for vibrational control. These include optical fibers, nitinol or constantin strain sensors, and PVDF or piezoceramics.
Piezoceramic sensors can be used as independent sensors or their functionality as sensor and actuator can be shared to form the sensoriactuator. In this embodiment, piezoceramic wafers serve as a collocated sensor and actuator. One advantage to smart structure sensors is the ability to spatially weight acoustically radiative modes by placing sensors in regions of high in-plane strain corresponding to the radiative mode.
Another advantage is the compactness of locating the sensor on the structure. A disadvantage is the necessity of formulating a relationship between a measurable structural parameter and the radiated acoustic pressure. This is only possible analytically for very few circumstances, as with the use of the Rayleigh integral to relate surface velocity to acoustic pressure when the structure is infinitely baffled. In the general case of a complex structure, this relationship between structural parameters and acoustic pressure is beyond the state of the art.
The determination of which modes are important as acoustic radiators and thus which modes to control, has been greatly simplified by the introduction of the wave-number transform, also called the k-transform. The k-transform is obtained by calculating the Fourier transform of a structure's spatial response. The resulting portion of the wavenumber spectrum below the wavenumber in the acoustic medium corresponds to the far-field radiation. The portion of the wavenumber spectrum above the wavenumber in the acoustic medium corresponds to the near-field radiation. This transform can be used to predict whether a vibrating structure will produce sound which propagates into the far field and to examine how changes introduced by active control will affect that propagation.
The majority of the active control approaches reviewed so far have been formulated in response to steady state sinusoidal disturbance inputs at one or multiple frequencies. The simplest control approach under these conditions is open-loop control. This can only be implemented when a very accurate representation of the disturbance signal can also be used to drive the actuators at a desired phase with respect to the radiating structure. The disadvantage of this approach is that it is not always possible to have a very accurate disturbance signal. A more sophisticated extension of this is the feedforward LMS adaptive approach. In this approach a quadratic cost function constructed of the acoustic error signals is minimized using superposed signals introduced by the actuator. An advantage of this approach is that it does not require a good estimate of the system and that it is relatively easy to implement in hardware. Smith, Fuller, and Burdisso found that for a broadband excitation, single-input-single-output (SISO) feedforward control did not give satisfactory performance in the attenuation of radiated sound from a plate. They found a multi-input-multi-output (MIMO) feedforward controller is necessary for significant acoustic attenuation. When the disturbance is broadband, a different approach is necessary for single-input-single-output systems. In order for the control to react quickly enough to the variable nature of the input, a feedback control approach must be formulated. Meirovitch and Thangjitham published one of the first theoretical studies using direct structural actuation and feedback control, but their approach was to minimize the vibration of a simply-supported elastic plate and to use the Rayleigh integral to check the effect of the control in the acoustic field. Also, they only attempted to control a harmonic disturbance. Bauman, Saunders, and Robertshaw used a Linear-Quadratic-Regulator (LQR) optimal method to suppress acoustic radiation from a beam that was excited by impulsive forces. They theorized that sound radiation from the beam would be suppressed by 73% with the controller configured to suppress vibration using LQR. Bauman, Ho, and Robertshaw also published a theoretical study investigating active acoustic control of broadband disturbances. Here, a feedback controller was designed for a clamped-clamped beam using a Linear-Quadratic-Gaussian (LQG) theory to minimize total radiated acoustic power.
The references all assumed direct structural actuation via an out-of-plane control force. There were also a few references found that investigated feedback control approaches using smart structural actuation. As was mentioned before, Banks and Fang described an acoustic cavity with one flexible beam boundary and smart structural actuation. Acoustic control was achieved using an LQR time domain approach, but the excitation was assumed to be periodic. Saunders, Cole, and Robertshaw examined stability criteria for collocated structural acoustic feedback control using sensoriactuators. They found that for partial state feedback of plant velocities and farfield radiation states, stability was not guaranteed, as is the case for direct velocity feedback in vibration control. Van Niekerk, Tongue, and Packard used an H.sub.2 optimal control procedure to design a dynamic feedforward/feedback controller to suppress transmission of a transient pulse through the previously described circular plate in a duct with piezoceramic actuators. Feedforward signals were provided by two microphones in the duct and a feedback signal was taken as the velocity of the center of the plate as measured by a laser vibrometer.
Among the acoustic control of sound transmission through flexible plates into three-dimensional cavities using smart structure actuation, Koshigoe, Gillis, and Falangas proposed a feedback method which makes the applied voltage to the piezoceramic proportional to sound pressure inside the cavity, but with the phase adjusted so as to create damping in the acoustic modes. They theorized that the method should be effective for both plate and cavity controlled modes. In the experimental study by Hill et al, several feedback control approaches including LQG/Loop Transfer Recovery (LTR), H_, pole placement and LQG were implemented based on the reduced order state space model, but the only input disturbance considered was harmonic.
A reasonable body of technical research exists for two popular acoustic instruments: the violin and the guitar. Both have been studied with respect to their structural/acoustic properties to some degree. The violin is considerably more complex than the guitar. The primary reasons for this are the asymmetrical vibration characteristics of the assembled violin and the involvement of the entire violin body in the production of sound. Despite the symmetrical shape, the bass bar and the soundpost located approximately on either side of the bridge below the top plate cause the vibration of the violin to be very complex and asymmetric. In fact, the primary purpose of the soundpost is to introduce asymmetry. It also effectively couples the top and back of the violin. Hutchins provides an extensive review of the history of violin research. In contrast, the sound radiated from the assembled guitar is primarily due to the vibration of the top plate which has lower frequency mode shapes that are relatively simple in comparison. As a result, the guitar is particularly amenable to modeling in its lower frequency function.
Of technical research that has been devoted to the modeling of acoustic-structural behavior of the acoustic guitar, most reported papers are concerned with the lower band of natural frequencies. This domain starts with the air mode at around 100 Hz and extends to the lowest plate mode of the lower bout of the acoustic guitar, which usually occurs around 200 Hz. Successful models of this low frequency behavior have drawn on an analogy to a vented loudspeaker enclosure with a solid piston representing the lower bout and an air piston representing the air mass that moves in and out of the rose. The pistons are constrained by an equivalent spring and damper whose parameters are derived from experimental measurements.
Firth described an analogous acoustical circuit used to model vented loudspeakers to describe the first two modes of the guitar. Frequency and damping parameters for this model were taken from admittance measurements made on a representative acoustic guitar. The analogous acoustical circuit was then used to predict pressure emanating from the guitar in the frequency range of the air mode and the first plate mode. These predictions were compared to measurements of sound output and its phase with relation to an excitation force at the center of the bridge. Extending this approach, Caldersmith used the analogy of a vented loudspeaker but derived the two coupled differential equations that describe the air mass that moves through the rose of the guitar as an air piston and the lower bout of the guitar as an equivalent plate piston. Stiffness and damping parameters for the pistons were taken from resonance and logarithmic decrement measurements, but an approach was outlined to estimate an equivalent stiffness for the plate piston directly for an assumed clamped orthotropic plate. SPL was calculated as a sum of the contribution of the air piston and the equivalent plate piston. Christensen and Vistisen used a similar approach but derived frequency and damping parameters entirely from top plate mobility measurements. A three-piston model has also been proposed by Christensen as an extension of the two-piston model that also treats the guitar back as an equivalent piston. Similar three-piston models were also described by Rossing, Popp, and Polstein and Fletcher and Rossing. Christensen also proposed modeling all top plate resonances up to 600 to 800 Hz as harmonically oscillating simple sources. This study included experimental measurements of resonant frequencies, initial guesses at damping and area to mass ratios and subsequent tuning of parameters to match experimental SPL measurements at one point in the acoustic field. It neglects multipole radiation of antisymmetric modes that could be significant in locations other than the measurement point two meters directly above the top plate. No published work could be found that links the spatial distribution of movement at the lower bout directly to the resulting sound pressure. This necessarily precludes consideration of sound pressure generated by antisymmetric plate modes at multiple locations in the acoustic field.
There are several factors in the low frequency regime of the acoustic guitar that have been identified as important in determining the quality of music the guitar is able to produce and, ultimately, the quality of the guitar itself. Specifically, these factors all are identifiable from structural transfer function measurements and SPL measurements made on the guitar. A study on appraisal of quality in guitars and violins was done by Gridnev and Porvenkov based on probabilistic spectrum analysis, but no specific advice on individual resonance properties was given. Christensen and Vistisen observed, based on a study of nine guitars, that the best guitars have the highest quality factors in their first resonance. They also observed that the lowest frequency should be relatively low.
By far the most thorough and conclusive research done on relating guitar quality to measurable quantities was by Meyer. In this work, 15 classical guitars of varying quality were used in a series of subjective and objective tests. The subjective tests consisted of a series of listening tests to different arrangements of music played on each guitar. The objective tests were performed by measuring frequency response characteristics in the SPL due to excitation of the guitars by an electrodynamic vibration system. Measurements were made using microphones in an anechoic chamber with the strings damped. Statistics were then employed to obtain a correlation between measured frequency response characteristics and subjective evaluations of the guitars. It was found that the three most highly correlated measurements with guitar quality were related to the antisymmetric mode of the guitar that occurs at around 400 Hz. This mode is also known as the (0,1) plate mode. Also, the factor with the highest negative correlation with quality was the quality factor in the air mode, meaning the air mode has high damping in guitars of high quality. Based on the results of the correlation tests, Meyer gives specific criteria for quality in acoustic guitars. Among these is the advice that the air mode and the first plate mode should have as much damping as possible, while the antisymmetric mode should have as little damping as possible. Also, the peak levels of the antisymmetric and first plate modes should be high.
Normally, advice on improving quality in guitars is directed at the skilled guitar luthier who achieves such changes passively by careful adjustments of thicknesses and bracing in the guitar. Christensen points out that strong excitation of the (0,1) antisymmetric plate mode is very difficult to achieve since the bridge is usually very close to its nodal line. The closer the bridge is located to the nodal line of a given mode, the less the excitation, of that mode, when the instrument is played.